Chapter 6
A probabilistic distribution function has two essential elements: the form of the function and the involved parameters. For example, the joint distribution of an MRF is characterized by a Gibbs function with a set of clique potential parameters; and the noise by a zero-mean Gaussian distribution parameterized by a variance. A probability model is incomplete if the involved parameters are not all specified even if the functional form of the distribution is known. While formulating the forms of objective functions such as the posterior distribution has long been a subject of research for in vision, estimating the involved parameters has a much shorter history. Generally, it is performed by optimizing a statistical criterion, e.g. using existing techniques such as maximum likelihood, coding, pseudo-likelihood, expectation-maximization, Bayes.
The problem of parameter estimation can have several levels of
complexity. The simplest is to estimate the parameters, denoted by
, of a single MRF, F, from the data d which is due to a
clean realization, f, of that MRF. Treatments are needed if the data
is noisy. When the noise parameters are unknown, they have to be
estimated, too, along with the MRF parameters. The complexity increases
when the given data is due to a realization of more than one MRF, e.g.
when multiple textures are present in the image data, and the data is
unsegmented. Since the parameters of an MRF have to be estimated from
the data, partitioning the data into distinct MRFs becomes a part of the
problem. The problem is even more complicated when the number of the
underlying MRFs is unknown and has to be determined. Furthermore, the
order of the neighborhood system and the largest size of cliques for a
Gibbs distribution can also be part of the parameters to be estimated.
The chief difficulty in ML estimation for MRF is the following: The
partition function Z in the Gibbs
distribution 
is also a function of 
and has
to be taken into consideration; since Z is calculated by summing over
all possible configurations, maximizing 
becomes
intractable, in general, even for small problems.
The example in Fig.6.1 illustrates the
significance of getting correct model parameters for MRF labeling
procedures to produce good results. The binary textures in row 1 are
generated using the MLL model (1.52) with the 
parameters being 
(for the left) and
(for the right), respectively. They have pixel
values of 100 and 160. Identical independently distributed Gaussian
noise 
is added to the pixel values, giving degraded images
in Row 2. Row 3 shows the result obtained using the true parameters of
the texture and the noise. Since such accurate information is usually
not available in practice, results can not be so good as that. Rows 4
and 5 show results obtained by using incorrect parameters
and 
, respectively.
These results are unacceptable.